MIMO (Space-Time Processing)ocw.snu.ac.kr/sites/default/files/NOTE/10358.pdf · 2018. 1. 30. · • S. H. Nam and K. B. Lee, “Transmit Power Allocation for an Extended V-BLAST

서울대학교 이동통신연구실 1

MIMO (Space-Time Processing)

서울대학교 이동통신연구실

• Fundamentals• V-Blast

• Improved V-Blast

• Simplified MIMO

• Detection schemes

• Performance in Correlated channel

• Detector implementations

• MIMO Channel capacity

• Waterfilling & SVD

• MIMO Capacity

Contents


Fundamentals

• Higher data rate is needed for next generation

communications in restricted bandwidth.

More spectrum efficient modulation technique

• Higher order modulation schemes: Vulnerable to noise

and interference.

• Multiple Input Multiple Output (MIMO) System

Using multiple antennas at Tx & Rx

Increase channel capacity and enhance performance without

bandwidth expansion.


11 12 131 1 1

2 21 22 23 2 2

3 3 331 32 33

y x n

y x n

y x n

y H x n

h h h

h h h

h h h

x1

x3

x2 y2

y3

y1h11

h31

h21 X

X

X


h11

h13

h12 X

X

X

g11

g13

g12+ x1

h12

h32

h22X

X

X

g11

g13

g12+ 0

21

31


If n=0,

11 12 13-1

21 22 23

31 32 33

g g g

H G g g g

g g g

-1 -1 ( )n

H y H Hx

x

11 12 13

21 22 23

31 32 33

-111 12 13

1 0 0

h h h

h h h

h h h

GH H H

g g g


V-Blast: Successive Interference Cancellation (MUD)

)(tr

)(1 tr

)(1 ts

)(2 tr

)(2 ts

)(1 tb )(2 tb

)(3 tr


Step 1. Order the transmitted signal

Step 2. Null the interference

Step 3. Detect the desired signal

Step 4. Cancel the detected signal from received

vector

V-BLAST (Ordered Successive Interference Cancellation)


< V-BLAST Receiver for 4 Tx-Antenna Systems >

LDor

MMSE

Decision for

Tx 1

NullingTx 2, 3, 4

-

1st Layer 2nd Layer

+Re-

Generation

-+

Re-Generation

-+

Re-Generation

LDor

MMSE

Decision for

Tx 3

NullingTx 4

LDor

MMSE

Decision for

Tx 2

NullingTx 3, 4

Decision for

Tx 4

3rd Layer 4th Layer

Combiner


• Shortcomings of V-BLAST receiver

- Diversity order

- Error propagation


• Overall BER (Assuming perfect symbol cancellation)

Vki: nulling vector at the ith stage

• Cost Function

• Differentiation of J w.r.t. transmit power

22

1 1

1 1( ) ( )

i

i

i

N Nk

b b k

i in k

PP e P e f

N N

v

1 2

1

( , , , ) ( )i

N

N b k

i

J P P P P e P N

ikP

22

, ( 1, 2,..., )i

ii

k

kn k

PdfN i N

dP

v

Improved V-Blast: Optimal TPA


Improved V-Blast: Effects of Detection Ordering and TPA

• Small diversity order for low detection stage Low detection stages dominate the overall performance

0 2 4 6 8 10 12 14 16 18 20 22 2410

-6

10-5

10-4

10-3

10-2

10-1

100

Stage 4

Stage 3

Stage 2

Stage 1

Without ordering, without TPA

BE

R

SNR per receive antenna [dB]



0 2 4 6 8 10 12 14 16 18 20 22 2410

-6

10-5

10-4

10-3

10-2

10-1

100

Stage 4

Stage 3

Stage 2

Stage 1


With ordering, without TPA

BE

R


• Detection ordering (1) shifts BER curves, and

(2) improves the 1st and 2nd stages, & degrades the 4th stage



0 2 4 6 8 10 12 14 16 18 20 22 2410

-6

10-5

10-4

10-3

10-2

10-1

100

Stage 4

Stage 3

Stage 2

Stage 1


With ordering, without TPA

With ordering, with TPA

BE

R


• TPA shifts the BER curves of the 1st and 2nd stages.


Improved V-Blast: Average Power of Optimal TPA

1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Avera

ge t

ran

sm

it p

ow

er

Detection stages

• Assign more transmit power to earlier detection stages

Compensation of low diversity orders at low stages


• Post-detection SNR for the kith Symbol

• BER for the kith Substream

• S. H. Nam and K. B. Lee, “Transmit Power Allocation for an Extended V-BLAST

System,” IEEE T-Comm, July 2004, pp 1074-1079.

22

i

i

i

k

k

n k

P

v

( ) ( )i ib k k

P e f

Improved V-Blast: Tx Power Allocat’n for min BER

1x

kx

Tnx

1p

kp

Tnp


Improved V-Blast: BER

• SNR gain at BER = 10-3: 4 dB (ZF), 2.5 dB (MMSE)

0 2 4 6 8 10 12 14 16 18 20 22 24 2610

-6

10-5

10-4

10-3

10-2

10-1

100

V-BLAST without TPA, ZF

V-BLAST with TPA, ZF

V-BLAST without TPA, MMSE

V-BLAST with TPA, MMSE

ML detection

BE

R



Detection Schemes

• V-Blast

• ML Receiver

• LD (Linear Decorrelator) Receiver

• MMSE Receiver


• ML Receiver

- Select the most likable transmitted vector.

- Complexity problem

• LD Receiver

- Null the interference signal using pseudo-inverse

matrix.

nHHHx

yHHHr

HH

HH

N

P 1

1


• MMSE Receiver

- minimize mean squared error due to interference signal

and noise.

1

2H H HP P EN N

r H H H n n y


< N=M=4 case > < N=M=6 case >

• BER Comparison between the Existed Schemes


- V-BLAST receivers outperforms MMSE or LD

receivers in terms of BER performance.

- Number of antennas

BER of LD

BER of MMSE …

BER of V-BLAST and ML

- Development of low complexity & high performance

receiver is needed.


• Channel Model

: Specular channel component : Scattered channel component (i.i.d.)

= Ricean Factor

T

sp r t

sc

a a

K

H

H

MIMO Channel Models


MIMO Performance in Correlated Channel

• Environments

• Channel Model

• QPSK Modulation

• SNR per Rx antenna


< N=M=4 case > < N=M=6 case >

• Average Capacity


Blank page


Posterior probabilities

signal was transmitted , 1,2, ,

After receiving the , the receiver choose that maximizes

Maximum a posterior probability (MAP)

: a prior probabili

m

m m

m m

m

m

P s r m M

r s p s r

f r s p sp s r

f r

p s

1

ty of the th signal.

M

m m

m

m

f r f r s p s

MIMO receiver 구현ML 설명 (7.5.3 The Optimum Detector, Proakis book)


2

2

0

10

2

0

10

2

1

1exp

1

2

Choose which maximizes

Choose which minimizes

N

N

m k mk

k

N

m k mk

k

m m

N

m k mk

k

f r s r s NN

Nln f r s ln N r s

N

s f r s

s r s

1 - signals are equally probable, .

- : independent of the transmitted signals.

- Choose which maximizes : ML criterion

: likelihood fn.

m

m m

m

M p sM

f r

s f r s

f r s

ML

MAP simplification

2

22

2

1: e

2

u mkr s

mf r s

참조


2 21 1 1

2 2

2

2

2

Choose which maximizes 2

: denote the region in the -dim space for which we decide

was transmitted when is received.

The probabilit

N N N

m k k mk mk

k k k

m m

m m m

m

m

D r s r r s s

r r s s

s r s s

R N

s r

y of a decision error given that was transmitted.

cm

m

m mR

s

P e s f r s dr

Prob. of error


1

1

1

1

11

is minimized by selecting if

for .

cm

M

m

m

M

mRm

M

R mm

m

m

m k

P e P e sM

f r s dr

f r s drM

P e s

f r s f r s m k

(M signals are equally probable.)

BPSKThreshold

s2 s1

QPSK


7.6 Probability of Error for Signal Detection in Additive White Gaussian Noise

2

0

2

0

1

1

0

2

0

1

1

b

b

b

r N

r N

r s n n

f r s eN

f r s eN


2

0

20

2

0

0

1 1

0

0

22

2

2

0

1

1

2

1

2

2

b

b

b

r N

Nx

x

N

b

P e s p r s dr

e drN

e dx

e dx

QN


Simplified Maximum Likelihood Detection

Scheme 1

• H. Z. Sung, J. W. Kang, and K. B. Lee, "A Simplified Maximum Likelihood Detection for

MIMO Systems," IEICE Transactions on Communications , vol. E98-B, no. 8, pp. 2241-2244,

Aug. 2006.


• Conventional ML detection scheme

– Performs likelihood test with all possible symbol

• Simplified ML detection scheme

– Step 1 & 2: Chooses candidate symbol combinations among all

possible symbol combinations

– Step 3: Performs likelihood test with candidate symbol

combinations

A Simplified ML Detection Scheme


- Rx signal: r

- Nulling

Nulling Matrix

Tentative statistic

- Tentative decision

The L closest elements of constellation point set to

-> L probable symbols for antenna symbol

11

2

:

:

T

n N

H H

M

y y y

P P

N N

y G r

G H H H I

G

y

,ˆ arg minn n

s S

x y s

• Step 1: Select probable symbols for each symbol

< Example: Probable

symbols >

(N=4 and L=2)

1,1x̂ 1,2x̂

2,2x̂

3,1x̂ 3,2x̂

4,1x̂ 4,2x̂

: Probable Symbols

2,1x̂

ny

nx


- Cancellation

- Nulling

where and

- Slicing & Constructing a candidate symbol combination

, ,ˆ

n n n

Px

N r r h

, , ,1 , , 1 ,

T

n n n N n ny y y G r

1

2H H

n n n n M

P P

N N

G H H H I 1 1 1 n n n N H h h h h

, , , ,ˆ ( ) 1, 2, , 1n i n ix Q y i N

,ˆ

n x , ,1ˆ[ nx , ,2ˆ nx , , 1ˆ n nx ,ˆ nx , ,ˆn nx , , 1ˆ ]T

n Nx

• Step 2: Determine a candidate symbol combination

for each probable symbol from step 1

< Example: Candidate symbol combinations >

(N=4 and L=2)

1,1x̂ 1,2x̂

2,2x̂

3,1x̂ 3,2x̂

4,1x̂ 4,2x̂

2,1x̂1,1,1x̂

1,1,2x̂

1,1,3x̂

1,2,1x̂

1,2,2x̂

1,2,3x̂

2,1,1x̂

2,1,2x̂

2,1,3x̂

2,2,1x̂

2,2,2x̂

2,2,3x̂

3,1,1x̂ 3,2,1x̂

3,1,2x̂

3,1,3x̂ 3,2,3x̂

3,2,2x̂

4,1,1x̂ 4,2,1x̂

4,1,2x̂ 4,2,2x̂

4,1,3x̂ 4,2,3x̂


- The likelihood function

- Decision value

,

1ˆexp ( ) ,

det( )nN

N

P

N

r H x

I

1,..., and 1,...L n N

,,

ˆˆarg min

nn

P

N

xr H x

,ˆ( )np r | x ,ˆ( )

H

n

P

N r H x

ˆ x

• Step 3. Determine the final decision value among

candidate symbol combinations

< Example: Decision value >

(N=4 and L=2)

2,2x̂

2,2,1x̂

2,2,2x̂

2,2,3x̂

2x̂

1̂x

3x̂

4x̂


BER versus average SNR: N=M=4, QPSK


• Example: Number of multiplications of the

proposed scheme and conventional detection

scheme for QPSK modulation (C=4)

N=M=4, L=4, and QPSK modulation: 72%

complexity reduction

The proposed schemeV-BLAST ML

L=1 L=2 L=3 L=4

N=M=4 1,008 1,122 1,296 1,440 467 5,120

N=M=6 4,632 5,100 5,568 6,036 1,955 172,032

Computational Complexity


Simplified Maximum Likelihood Detection

Scheme 2

• J. W. Kang and K. B. Lee, "A Multi-stage ML Detection for MIMO Systems," conditionally

accepted to IEICE Transactions on Communications in May 2005.


* The number of surviving symbols; L = 2

1x

2x

3x

4x

Interfering sub-streams

: Objective sub-stream

1x

2x

3x

4x


1,1s

1,4s1,2s 1,3s

: Candidate symbols

1x

2x

3x

4x


1,1s

1,4s1,2s 1,3s

: Candidate symbols

Perform likelihood test with 1 1,1 1,2 1,3 1,4, , , S s s s s

< The 1st stage >

1x

2x

3x

4x


1,1y 1,2y

: Candidate symbols

: Surviving symbols

* C candidate symbols L surviving symbols


* CL candidate symbol combinations

L surviving symbol combinations

1x

2x

3x

4x


1x

2x

3x

4x


2,1s

2,2s 2,5s 2,6s 2,7s 2,8s

2,3s 2,4s

1x

2x

3x

4x


2,1s

2,2s 2,5s 2,6s 2,7s 2,8s

2,1y 2,2y

Perform likelihood test with 2 2,1 2,2 2,3 2,8, , , ... ,S s s s s

1x

2x

3x

4x


< The 2nd stage >


BER versus average SNR (N=M=4), QPSK


Capacity in band-limited, power-limited

gaussian channel

•

•

•

Y X Nk k k

C I X Y X E X Pk k k k ( ; ) : [ ] Gaussian, 2

CP

1

212 2log ( )


MIMO Channel Capacity

• Capacity per Hz

• Independent data stream, equal power allocation

– CSI not available in Tx

2 2log det H H

M X X

PC E

N

I H K H K XX

2 2log det HM

PC

N

I HH


• At large n and high SNR

Capacity grows linearly with the number of antenna

e

PnC

22log


Water Filling Algorithm in Parallel Channels

• K independent channels in parallel

– independent Gaussian noise for each channel

• Constraint on total transmit power

Y3

Y2

Y1

Z1

X2

X1

Z2

Z3

X3

h1

h3

h2

, 1,2,...,

~ (0, )

j j j j

j j

Y h X Z j k

Z N

N

E X Pjj

k2

1


Parallel Gaussian Channel (Cont.)

• It’s achieved when

• Maximize capacity using Lagrange multipliers

• Differentiating with respect to

CP

NEX P P

i

k

i

i

i i

1

21

1

2log( ) , where P i

( , ,..., ) ~X X X k1 2

0 0

0 0

0 0

N 0,

P

P

P

1

2

k

L

N

MMMM

O

Q

PPPP

F

H

GGGG

I

K

JJJJ

J P P PP

NP Pk

i

i

i( , ,..., ) log( ) ( )1 21

21

Pi


Parallel Gaussian Channel (Cont.)

• Power must be non-negative

1

2

10

1

2

1

2

P N

P N

P N

i i

i i

i i

FHG

IKJ

,

P Ni i ( ) +

( ) N Pi

Power

Channel 1 Channel 3Channel 2

P2

N2

N3

N1

P1


Water Filling & SVD for max capacity

• By decoupling transformation, MIMO channel is

transformed into parallel SISO channel

•

•

D : eigenvalues of HH2 HH UDV H ,

Y X N Y X N U HV D UH H( )

V

Decoupling

Transform

Tnp

1p

kp

1s

ks

Tns

H

UH

Decoupling

Transform

Tn Rn

1

~S

kS~

TnS~

Documents

MIMO (Space-Time Processing)ocw.snu.ac.kr/sites/default/files/NOTE/10358.pdf · 2018. 1. 30. · • S. H. Nam and K. B. Lee, “Transmit Power Allocation for an Extended V-BLAST